To solve the equation x² – 14x + 17 = 96, we first need to rearrange it into standard form. We can do this by moving 96 to the left side of the equation:
x² – 14x + 17 – 96 = 0
This simplifies to:
x² – 14x – 79 = 0
Now we have a quadratic equation in the form of ax² + bx + c = 0, where:
- a = 1
- b = -14
- c = -79
We can solve for x using the quadratic formula, which is:
x = (-b ± √(b² – 4ac)) / (2a)
Let’s substitute our values into the formula:
x = (14 ± √((-14)² – 4(1)(-79))) / (2(1))
This further simplifies to:
x = (14 ± √(196 + 316)) / 2
Calculating the inside of the square root:
x = (14 ± √(512)) / 2
The square root of 512 can be simplified to:
√512 = √(256 * 2) = 16√2
Now substituting back, we have:
x = (14 ± 16√2) / 2
Breaking it down:
x = 7 ± 8√2
So, the solutions for x are:
x₁ = 7 + 8√2 and x₂ = 7 – 8√2